I'm sorry to hear that your periods are so difficult for you. Menstrual suppression is definitely one way to help with any distress around periods and gender.
Hormonal IUDs, like Mirena and Skyla, can make your periods lighter, or even make them go away (though we can't guarantee that this will happen). It's important to note that copper IUDs will not
make your period go away, and may even make them heavier, so for this purpose it will be important to ask your provider for counseling around a hormonal IUD specifically.
We have some more information on IUDs here
. Many people do experience discomfort or pain during and immediately after insertion. Tylenol and heating pads should help with this pain.
There are other methods that can work for menstrual suppression, too. If you'd prefer to try something else in the short-term before committing to a long-acting reversible contraceptive, like the IUD, you can try using birth control pills or the NuvaRing to suppress your periods. Note that the NuvaRing will require you to insert something into your vagina about once a month, though, so if that will be emotionally or physically difficult, I would recommend birth control pills or the IUD.
I want to say that you should never pull your IUD out yourself -- only a medical professional can remove this device properly and safely. It is easy to forget your IUD is even there after a while, because you don't really feel it, so I'm hoping that might help with the urges you described. Do you want to say more about why you might remove or cease birth control because of gender feelings? I want to make sure we support you in interrogating and diminishing that desire, if we can. <3